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ESAT formulas and facts

The ESAT gives you no formula booklet and no calculator, so everything below has to come from memory and mental arithmetic. This is what is worth knowing cold, module by module, with the shortcuts that recur most often and the traps that cost the most marks.

The pacing arithmetic, which decides more marks than any formula

Each module is 27 multiple choice questions in 40 minutes. That is about 89 seconds a question, and that average already includes the time you spend reading.

So you cannot afford a full derivation on every question. Budget around 60 seconds for the ones you recognise, which buys two to three minutes for the handful that are genuinely hard. If a question has taken two minutes with no route in sight, mark it and move on. There is no negative marking, so an unanswered question is strictly worse than a guess.

Read the options before you start solving. They tell you the form the answer has to take, exact surd or decimal, units, sign, and they often let you discard two or three on magnitude alone, sometimes with no algebra at all.

Three habits that pay across every module

Square and cube your scale factors. If lengths scale by k, areas scale by k² and volumes by k³. Doubling every length makes an area four times as large and a volume eight times. That one idea accounts for a large share of the geometry.
Filter roots against physical reality. The algebra may hand you three roots, but lengths, masses and radii have to be positive, so the rest go. A question whose answer is "no valid values" is testing exactly this.
Check what was actually asked. Having found x, confirm the question did not ask for x³, or for RS = x + 5. This is the single most common way to lose a mark you had already earned.

Mathematics

Circle and sector. A = πr², C = 2πr. Sector = ½r²θ; segment = ½r²(θ − sin θ), with θ in radians.
Clock geometry. The minute hand moves 6° a minute, the hour hand 0.5° a minute, so they close at 5.5° a minute. Each hour mark is 30°.
Surds, indices and standard form. Match the powers of ten before adding numbers in standard form.
Counting. Keep the "+1" in an inclusive count: from 7 to 19 inclusive is 13 numbers, not 12.

The slips that recur: forgetting the plus or minus when you take a square root, dropping the "+1" in an inclusive count, and adding standard form numbers without matching the powers of ten first.

Mathematics 2

Stationary points. Set f′(x) = 0. Then f″ > 0 is a minimum and f″ < 0 a maximum. For a cubic whose f′ has roots x₁ and x₂, f′(x) = k(x − x₁)(x − x₂).
Trigonometric identities. sin 2x = 2 sin x cos x; cos 2x = 2 cos²x − 1; sin²x + cos²x = 1. For a multiple angle, scale the domain before you list the solutions.
Polynomials. Test factors of the constant term over factors of the leading coefficient. (x − y) always divides xⁿ − yⁿ. After extracting a root, check the discriminant of what is left.
Logic. Only the contrapositive, not Q ⇒ not P, is equivalent to P ⇒ Q. Concluding the converse or the inverse is a fallacy, and a counterexample has to satisfy the premise and fail the conclusion.
Recurrence relations. If one looks intractable, compute the first four or five terms. It is very often periodic.

The method that deletes half a definite integral

Physics

Kinematics. v = u + at; s = ut + ½at²; v² = u² + 2as. On a velocity against time graph the gradient is acceleration and the signed area is displacement.
Momentum. p = mv, conserved in any collision with no external force; kinetic energy only in elastic ones. Impulse is FΔt = mΔv, and on a bounce Δv is the sum of the two speeds, not the difference.
Springs behave oppositely to resistors. F = kx. Two equal resistors give 2R in series and R/2 in parallel; two equal springs give k/2 in series and 2k in parallel.
Waves. v = fλ and T = 1/f. A particle travels 4A per cycle whatever the wavelength, and compression to rarefaction is half a period.

Drag and sliding friction always oppose the current direction of motion, so the force equation flips sign the moment an object reverses. That is where terminal velocity questions catch people.

The method that cancels every constant

Chemistry

Le Chatelier. The system shifts to oppose the change. Higher pressure favours the side with fewer gas moles, and heating favours the endothermic direction. A catalyst changes the rate only, never the position or the yield.
Energetics. q = mcΔT. Reversing an equation flips the sign of ΔH, and multiplying it scales ΔH.
Redox. Oxidation is loss of electrons. Disproportionation, one element both oxidised and reduced, is always a redox reaction. Balance by equating the electrons transferred.
Electron configuration. Shells fill 2, 8, 8, 2 up to calcium, Z = 20. Past that, 4s fills before 3d.
Organic. Alkene with HBr is electrophilic addition, and Markovnikov puts the halogen on the carbon giving the more stable carbocation, tertiary over secondary over primary. The carbon to fluorine bond is the strongest carbon to halogen bond, so it is the least reactive.

Biology

Hardy Weinberg. p + q = 1 and p² + 2pq + q² = 1. Homozygous recessives are q² and heterozygotes 2pq; multiply by the population for counts.
Genetics. Aa × Aa gives 3:1 by phenotype and 1:2:1 by genotype. Two recessive parents can never produce a dominant offspring, which is the fastest elimination there is on a pedigree.
DNA. A pairs with T by two hydrogen bonds, C with G by three, so A = T and C = G. Insertions and deletions cause a frameshift unless the number of bases is a multiple of three; substitutions may be silent.
Transport. Diffusion and osmosis are passive and go down a gradient. Anything moving against a gradient, or moved in bulk by endocytosis or exocytosis, needs ATP and therefore a living cell.
Enzymes. Rate rises to an optimum then falls sharply as the enzyme denatures. More substrate overcomes a competitive inhibitor. Pepsin works at pH 1.5 to 2, intestinal proteases near pH 8.
Ecology. Roughly a tenth of the energy transfers between trophic levels; the rest is lost as heat from respiration, in the parts not eaten, and in waste. Kidneys remove metabolic waste from the blood, and faeces are digestive waste that never entered it.

The traps these papers set most often

Extraneous data. Wall heights, wavelengths and masses are routinely supplied and not needed. An unused number is not a signal that you have missed a step.
Round trips. In echo, sonar and ultrasound questions the pulse travels twice the depth, so depth = ½vt.
Energy barriers. Before computing a final speed from a drop in height, check the object clears any intermediate peak at all.
Domain restrictions. Cancelling a factor from an algebraic fraction can silently introduce a root, and dividing by a factor that may be zero can silently lose one. State the excluded values first, then check your answers against them.
Order of operations. Scan every option for multiplication and brackets before evaluating anything left to right.
"None of the above" is a real answer. If your derivation is sound and the value is not there, trust the derivation.

How to use this page

Reading it through once is worth little. The loop that works is to sit a module under timed conditions, then come back and re-read only the rows that cost you marks. That is also the fastest way to find out which of these you only think you know.

Sit a timed module

What this page is for

It is written for a candidate preparing on their own, and it is free to read with no account. It is not licensed for a school, college or tutoring provider to print and run with a class, to republish, or to fold into a course sold by another provider. A school that wants a paper written for its own candidates and issued to nobody else wants a commissioned pack.